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<span class="kw">let </span><font color="Maroon" title="c1">CNS</font> be   <a href="clvect_1.html#NM5" title="CLVECT_1:NM.5">ComplexNormSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">RNS</font> being  <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a><br/> for <font color="Olive" title="b2">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b3">f</font> being  <a href="struct_0.html#NM10" title="STRUCT_0:NM.10">PartFunc</a> of <font color="Olive" title="b1">RNS</font>,<font color="Maroon" title="c1">CNS</font>  st <font color="Olive" title="b3">f</font> <a href="ncfcont1.html#R15" title="NCFCONT1:pred.15">is_Lipschitzian_on</a> <font color="Olive" title="b2">X</font> holds <br/>(  <a href="vfunct_2.html#K6" title="VFUNCT_2:func.6">-</a> <font color="Olive" title="b3">f</font> <a href="ncfcont1.html#R15" title="NCFCONT1:pred.15">is_Lipschitzian_on</a> <font color="Olive" title="b2">X</font> &amp; <span class="p1"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Olive" title="b3">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span> <a href="nfcont_1.html#R6" title="NFCONT_1:pred.6">is_Lipschitzian_on</a> <font color="Olive" title="b2">X</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">RNS</font> be   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">f</font> being  <a href="struct_0.html#NM10" title="STRUCT_0:NM.10">PartFunc</a> of <font color="Maroon" title="c2">RNS</font>,<font color="Maroon" title="c1">CNS</font>  st <font color="Olive" title="b2">f</font> <a href="ncfcont1.html#R15" title="NCFCONT1:pred.15">is_Lipschitzian_on</a> <font color="Olive" title="b1">X</font> holds <br/>(  <a href="vfunct_2.html#K6" title="VFUNCT_2:func.6">-</a> <font color="Olive" title="b2">f</font> <a href="ncfcont1.html#R15" title="NCFCONT1:pred.15">is_Lipschitzian_on</a> <font color="Olive" title="b1">X</font> &amp; <span class="p1"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Olive" title="b2">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span> <a href="nfcont_1.html#R6" title="NFCONT_1:pred.6">is_Lipschitzian_on</a> <font color="Olive" title="b1">X</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="struct_0.html#NM10" title="STRUCT_0:NM.10">PartFunc</a> of <font color="Maroon" title="c2">RNS</font>,<font color="Maroon" title="c1">CNS</font>  st <font color="Olive" title="b1">f</font> <a href="ncfcont1.html#R15" title="NCFCONT1:pred.15">is_Lipschitzian_on</a> <font color="Maroon" title="c3">X</font> holds <br/>(  <a href="vfunct_2.html#K6" title="VFUNCT_2:func.6">-</a> <font color="Olive" title="b1">f</font> <a href="ncfcont1.html#R15" title="NCFCONT1:pred.15">is_Lipschitzian_on</a> <font color="Maroon" title="c3">X</font> &amp; <span class="p1"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Olive" title="b1">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span> <a href="nfcont_1.html#R6" title="NFCONT_1:pred.6">is_Lipschitzian_on</a> <font color="Maroon" title="c3">X</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">f</font> be   <a href="struct_0.html#NM10" title="STRUCT_0:NM.10">PartFunc</a> of <font color="Maroon" title="c2">RNS</font>,<font color="Maroon" title="c1">CNS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">f</font> <a href="ncfcont1.html#R15" title="NCFCONT1:pred.15">is_Lipschitzian_on</a> <font color="Maroon" title="c3">X</font> implies (  <a href="vfunct_2.html#K6" title="VFUNCT_2:func.6">-</a> <font color="Maroon" title="c4">f</font> <a href="ncfcont1.html#R15" title="NCFCONT1:pred.15">is_Lipschitzian_on</a> <font color="Maroon" title="c3">X</font> &amp; <span class="p1"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span> <a href="nfcont_1.html#R6" title="NFCONT_1:pred.6">is_Lipschitzian_on</a> <font color="Maroon" title="c3">X</font> ) )</span><br/>







<span class="kw">assume </span><a NAME="E1:156"/><span class="lab"><font color="Green" title="E91">A1</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="ncfcont1.html#R15" title="NCFCONT1:pred.15">is_Lipschitzian_on</a> <font color="Maroon" title="c3">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="vfunct_2.html#K6" title="VFUNCT_2:func.6">-</a> <font color="Maroon" title="c4">f</font> <a href="ncfcont1.html#R15" title="NCFCONT1:pred.15">is_Lipschitzian_on</a> <font color="Maroon" title="c3">X</font> &amp; <span class="p1"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span> <a href="nfcont_1.html#R6" title="NFCONT_1:pred.6">is_Lipschitzian_on</a> <font color="Maroon" title="c3">X</font> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">s</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E3:156"/><span class="lab"><font color="Green" title="E92">A2</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">s</font>
 <span class="kw">and </span><br/><a NAME="E4:156"/><span class="lab"><font color="Green" title="E93">A3</font></span>: 
for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">RNS</font>  st <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Olive" title="b2">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> holds <br/><span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x1</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">x2</font></span>)</span></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">s</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">x1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Olive" title="b2">x2</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ncfcont1.html#D29" target="_self" title="NCFCONT1:def.29">Def29</a></span>;<br/>
<a NAME="E5:156"/>
 <a href="vfunct_2.html#K6" title="VFUNCT_2:func.6">-</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a> </span>)</span> <a href="vfunct_2.html#K4" title="VFUNCT_2:func.4">(#)</a> <font color="Maroon" title="c4">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vfunct_2.html#T23" title="VFUNCT_2:th.23">VFUNCT_2:23</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:156"/>
 <a href="vfunct_2.html#K6" title="VFUNCT_2:func.6">-</a> <font color="Maroon" title="c4">f</font> <a href="ncfcont1.html#R15" title="NCFCONT1:pred.15">is_Lipschitzian_on</a> <font color="Maroon" title="c3">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:156"><span class="lab"><font color="Green" title="E91">A1</font></span></a>, <a class="ref" href="ncfcont1.html#T129" target="_self" title="NCFCONT1:th.129">Th129</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span> <a href="nfcont_1.html#R6" title="NFCONT_1:pred.6">is_Lipschitzian_on</a> <font color="Maroon" title="c3">X</font></span><br/>

<a NAME="E7:156"/>
<font color="Maroon" title="c3">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:156"><span class="lab"><font color="Green" title="E91">A1</font></span></a>, <a class="ref" href="ncfcont1.html#D29" target="_self" title="NCFCONT1:def.29">Def29</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:156"/><span class="lab"><font color="Green" title="E94">A4</font></span>: 
<font color="Maroon" title="c3">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ncfcont1.html#D4" target="_self" title="NCFCONT1:def.4">Def4</a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="nfcont_1.html#D14" title="NFCONT_1:def.14">NFCONT_1:def 14</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>( not <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">RNS</font>  holds <br/>(  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> or  not <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> or  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>3</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Olive">b<sub>3</sub></font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> ) ) )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c5">s</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( not <font color="Maroon" title="c5">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">RNS</font>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> or  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> or  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">s</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E9:156"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:156"><span class="lab"><font color="Green" title="E92">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">RNS</font>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> or  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> or  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">s</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c6">x1</font>, <font color="Maroon" title="c7">x2</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">RNS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> or  not <font color="Maroon" title="c7">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> or  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">x1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">x2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">s</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c7">x2</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E10:156"/><span class="lab"><font color="Green" title="E95">A5</font></span>: 
<font color="Maroon" title="c6">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 <span class="kw">and </span><br/><a NAME="E11:156"/><span class="lab"><font color="Green" title="E96">A6</font></span>: 
<font color="Maroon" title="c7">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">x1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">x2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">s</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c7">x2</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span></span><br/>

 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">x1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">x2</font></span>)</span></span>)</span> = 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">x2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:156"><span class="lab"><font color="Green" title="E94">A4</font></span></a>, <a class="txt" href="#E10:156"><span class="lab"><font color="Green" title="E95">A5</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>
<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">x2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:156"><span class="lab"><font color="Green" title="E94">A4</font></span></a>, <a class="txt" href="#E11:156"><span class="lab"><font color="Green" title="E96">A6</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>
<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">x1</font></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">x2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:156"><span class="lab"><font color="Green" title="E94">A4</font></span></a>, <a class="txt" href="#E10:156"><span class="lab"><font color="Green" title="E95">A5</font></span></a>, <a class="ref" href="ncfcont1.html#D4" target="_self" title="NCFCONT1:def.4">Def4</a></span>
<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">x1</font></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">x2</font></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:156"><span class="lab"><font color="Green" title="E94">A4</font></span></a>, <a class="txt" href="#E11:156"><span class="lab"><font color="Green" title="E96">A6</font></span></a>, <a class="ref" href="ncfcont1.html#D4" target="_self" title="NCFCONT1:def.4">Def4</a></span>
;<br/>
<a NAME="E13:156"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A7</font></span>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">x1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">x2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">x1</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">x2</font></span>)</span></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="clvect_1.html#T111" title="CLVECT_1:th.111">CLVECT_1:111</a></span>;<br/>
<a NAME="E14:156"/>
<span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">x1</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">x2</font></span>)</span></span>)</span></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">s</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c7">x2</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:156"><span class="lab"><font color="Green" title="E93">A3</font></span></a>, <a class="txt" href="#E10:156"><span class="lab"><font color="Green" title="E95">A5</font></span></a>, <a class="txt" href="#E11:156"><span class="lab"><font color="Green" title="E96">A6</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:156"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">x1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">x2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">s</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c7">x2</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:156"><span class="lab"><font color="Green" title="E97">A7</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
